The Pentagon has warned that targeted American airstrikes won’t be enough to roll back the bloody advance of ISIL Takfiri terrorists in Iraq.

American strikes have only “temporarily disrupted” the extremist group’s shockingly effective onslaught, the Pentagon’s Lieutenant General William Mayville told reporters on Monday.

“I in no way want to suggest that we have effectively contained, or that we are somehow breaking the momentum of, the threat posed by ISIL,” he said.

The comments came as leading Republican congressmen asked President Barack Obama to step up US military operation in Iraq, warning that the limited airstrikes the president authorized this week will not be enough to turn the tide against the militants.

"Launching three strikes around a place where a horrible humanitarian crisis is taking place...is clearly very, very ineffective, to say the least," Republican Senator John McCain said.

The hawkish Republican counseled the president to expand the airstrikes to neighboring Syria, where ISIL has gained a significant foothold, saying the border between the two countries had effectively been "erased."

He also pressed for increased assistance to Iraqi Kurds, who are struggling to fend off the ISIL advance, and urged Obama not to wait for political reconciliation in Baghdad before expanding America's involvement in the conflict.

Republican Sen. Lindsey Graham from South California concurred with McCain about the president's lack of a strategy, saying much of Obama's decision-making thus far seems designed to "avoid a bad news story on his watch."

Republican Peter King, expressed a similar concern urged the president to "take nothing off the table" in securing American interests.

Several former military and diplomatic officials, though, said America's airstrikes would suffice to contain ISIL, but they likely wouldn't be enough to put the group on a path to defeat.

ISIL "can be stopped by a combination of people on the ground who are willing to fight, such as the Peshmerga, and American airstrikes such as we have seen," said former US Ambassador to Iraq James Jeffrey on CBS News.

Retired General Carter Ham, a former commander of the US forces in Mosul, said the initial American airstrikes "are already having some effect."

Democrats, too, said the air strikes are beginning to take a toll on ISIL, and they applauded the president for strictly limiting the mission in Iraq, saying Americans simply don't have the appetite for more sustained military engagement.

"The basic strategy is targeted strikes on these weapons systems so that the Kurdish Peshmerga forces can strengthen and resist and ultimately roll back ISIL”. Senator Jack Reed told CBS.

US bombers have carried out airstrikes on the ISIL militants’ positions since Friday helping the Kurdish forces to retake the towns of Makhmour and al-Gweir, some 28 miles (45 kilometers) from Erbil, north of the Iraq, from the Takfiri militants.

ISIL militants have terrorized entire communities, including Shias, Sunnis, Kurds, Christians, Yazidis and others, as they continue their advances in Iraq.

According to UN reports, more than one million Iraqis have been displaced as the result of the ISIL attacks in northern Iraq.